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On my team there is a desire for some users to periodically update their MySQL password.

Our MySQL server is shared between a few projects (and certain users are assigned to certain projects). If I'm correct, changing your password requires access to the mysql.user table (which would, in turn allow the user to modify the password of any other user on the server.)

If this is all correct, is there any way to secure the database in a way that a user can update their password, but not other users?

I guess this is not handled in MySQL, infact root user only should give access to other users and he only should change their password if needs to be ..IMHO
technically in order to change password you need to access mysql.user table.